Timeline for Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 14, 2015 at 18:37 | comment | added | Eric Wofsey | More generally, both $K_1$ and $K_2$ must have the property that the Hurewicz map $\pi_2\to H_2$ vanishes, because any inclusion of 2-dimensional complexes is injective on $H_2$. | |
| Jan 14, 2015 at 15:21 | comment | added | Samarkand | I have got almost the same answer on MSE (it was removed), of course, as Andre noticed it cannot work. | |
| Jan 14, 2015 at 10:17 | comment | added | André Henriques | Indeed, this cannot work: there is no embeding from $RP^2\cup C$ into any 2-dimensional CW complex $Y$ such that the induced map $\pi_2(RP^2\cup C)\to\pi_2(Y)$ is zero. Easier example: there is no embeding from $S^2$ into any 2-dimensional CW complex $Y$ such that the induced map $\pi_2(S^2)\to\pi_2(Y)$ is zero. The reason is that any such embedding $S^2\to Y$ admits a retract $Y\to S^2$ up to homotopy. | |
| Jan 14, 2015 at 8:56 | comment | added | Grigory M | Are you sure $K_2\to K_3$ induces a zero map on $\pi_2$? Why? | |
| Jan 14, 2015 at 8:01 | history | answered | user43326 | CC BY-SA 3.0 |